Class summary:
Friday, 3/14
[This material will not be on exams.]
I discussed a variation of a famous paradox, known as the "St. Petersbug
paradox" (or "St. Petersburg game"). It involves playing a game with
even odds (say, tossing a coin),
in which you win or lose $1 (net) for each bet of $1,
depending whether you win or lose the game. You are free to set the
amount to bet on the game, and to stop playing at any time.
With most naive strategies, like betting $1 on each game, your expected
net win after a series of games is $0. However,
it turns out that there is a strategy that guarantees you a $1 net
win eventually, no matter how the individual games turn out; hence
your expected net win under this strategy is $1.
Namely, you start betting $1 on the first game, and then keep doubling
your bet until you win a game, at which point you stop.
It is not hard to see that (a) after the win occurs (and you therefore
stop) you are
ahead by exactly $1, and (b) that the probability that you win a game
at some point is 1. Thus, the strategy guarantees you a win
with 100 percent probability.
The usual formulation of the St Petersburg paradox is somewhat
different, but based on the same idea: It involves a casino game with
even odds, and you keep playing this game until you win. At that
point, the casino pays you $2n, where n is the number of
games you have played. One can show that the expected win is infinite.
Hence, you matter how much the casino charges you to play this game,
you will come out ahead.
Of course, there are a few catches, that prevent one from exploiting
this game in the real world. In particular, the amount needed to bet
under the "double-if-you-lose"
strategy becomes astronomical after only a few dozen games, so in
reality you cannot play arbitrarily many games. However, if the number
of games is restricted (say, at most 20 games so that the bet does not
exceed $220), then the paradox disappears: the expected
net win under such a restriction is 0.
Here are two websites dealing with this paradox:
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